A basis is a linearly independent list that also spans V.
Some Theorems and Counter Examples
2.28 A list of vectors isa basis if every vector in V can be written as a unique linear combination of vectors in the basis.
2.30 Every spanning list in a vector space can be reduced to a basis
2.31 Every finite dimensional vector space has a basis
2.32. Every linearly independent list extends to a basis
2.33 Every subspace of V is part of a direct sum equal to V
2C Dimension
Any two bases of a finite dimensional vector space have the same length.
Dimension of a finite-dimensional vector space is the length of any basis of that space.
Some Theorems and Counterexamples
2.37 If V is finite dimensional and U is a subspace of V, then dim U <= dim V.
2.38 Every linearly independent list of vectors of V of length dim V is a basis.
2.39 A subspace of full dimension equals the whole space.
2.42 Every spanning list of vectors in V of length dim V is a basis of V.
2.43 dim(v1 + v2) = dim v1 + dim v2 -dim(v1 intersect v2)